Redshift-space Galaxy Research Articles

Cosmological parameters and neutrino masses from the final Planck and full-shape BOSS data

We present a joint analysis of the Planck cosmic microwave background (CMB) and Baryon Oscillation Spectroscopic Survey (BOSS) final data releases. A key novelty of our study is the use of a new full-shape (FS) likelihood for the redshift-space galaxy power spectrum of the BOSS data, based on an improved perturbation theory template. We show that the addition of the redshift space galaxy clustering measurements breaks degeneracies present in the CMB data alone and tightens constraints on cosmological parameters. Assuming the minimal $\Lambda$CDM cosmology with massive neutrinos, we find the following late-Universe parameters: the Hubble constant \mbox{$H_0=67.95^{+0.66}_{-0.52}$ km s$^{-1}$Mpc$^{-1}$}, the matter density fraction \mbox{$\Omega_m=0.3079^{+0.0065}_{-0.0085}\,$}, the mass fluctuation amplitude \mbox{$\sigma_8=0.8087_{-0.0072}^{+0.012}\,$}, and an upper limit on the sum of neutrino masses \mbox{$M_{\text{tot}} <0.16\,$ eV} ($95\%$ CL).This can be contrasted with the Planck-only measurements: \mbox{$H_0=67.14_{-0.72}^{+1.3}$} km s$^{-1}$Mpc$^{-1}$, $\Omega_m=0.3188^{+0.0091}_{-0.016}\,$, \mbox{$\sigma_8=0.8053_{-0.0091}^{+0.019}\,$}, and \mbox{$M_{\text{tot}} <0.26\,$ eV} ($95\%$ CL). Our bound on the sum of neutrino masses relaxes once the hierarchy-dependent priors from the oscillation experiments are imposed. The addition of the new FS likelihood also constrains the effective number of extra relativistic degrees of freedom, \mbox{$N_{\text{eff}}=2.88\pm 0.17$}. Our study shows that the current FS and the pure baryon acoustic oscillation data add a similar amount of information in combination with the Planck likelihood. We argue that this is just a coincidence given the BOSS volume and efficiency of the current reconstruction algorithms.In the era of future surveys FS will play a dominant role in cosmological parameter measurements.

Producing a BOSS CMASS sample with DES imaging

ABSTRACT We present a sample of galaxies with the Dark Energy Survey (DES) photometry that replicates the properties of the BOSS CMASS sample. The CMASS galaxy sample has been well characterized by the Sloan Digital Sky Survey (SDSS) collaboration and was used to obtain the most powerful redshift-space galaxy clustering measurements to date. A joint analysis of redshift-space distortions (such as those probed by CMASS from SDSS) and a galaxy–galaxy lensing measurement for an equivalent sample from DES can provide powerful cosmological constraints. Unfortunately, the DES and SDSS-BOSS footprints have only minimal overlap, primarily on the celestial equator near the SDSS Stripe 82 region. Using this overlap, we build a robust Bayesian model to select CMASS-like galaxies in the remainder of the DES footprint. The newly defined DES-CMASS (DMASS) sample consists of 117 293 effective galaxies covering $1244\,\deg ^2$. Through various validation tests, we show that the DMASS sample selected by this model matches well with the BOSS CMASS sample, specifically in the South Galactic cap (SGC) region that includes Stripe 82. Combining measurements of the angular correlation function and the clustering-z distribution of DMASS, we constrain the difference in mean galaxy bias and mean redshift between the BOSS CMASS and DMASS samples to be $\Delta b = 0.010^{+0.045}_{-0.052}$ and $\Delta z = \left(3.46^{+5.48}_{-5.55} \right) \times 10^{-3}$ for the SGC portion of CMASS, and $\Delta b = 0.044^{+0.044}_{-0.043}$ and $\Delta z= (3.51^{+4.93}_{-5.91}) \times 10^{-3}$ for the full CMASS sample. These values indicate that the mean bias of galaxies and mean redshift in the DMASS sample are consistent with both CMASS samples within 1σ.

Alcock–Paczynski Test with the Evolution of Redshift-space Galaxy Clustering Anisotropy

Abstract We develop an improved Alcock–Paczynski (AP) test method that uses the redshift-space two-point correlation function (2pCF) of galaxies. Cosmological constraints can be obtained by examining the redshift dependence of the normalized 2pCF, which should not change apart from the expected small nonlinear evolution. An incorrect choice of cosmology used to convert redshift to comoving distance will manifest itself as redshift-dependent 2pCF. Our method decomposes the redshift difference of the two-dimensional correlation function into the Legendre polynomials whose amplitudes are modeled by radial fitting functions. Our likelihood analysis with this 2D fitting scheme tightens the constraints on Ω m and w by ∼40% compared to the method of Li et al. that uses one-dimensional angular dependence only. We also find that the correction for the nonlinear evolution in the 2pCF has a non-negligible cosmology dependence, which has been neglected in previous similar studies by Li et al. With an accurate accounting for the nonlinear systematics and use of full two-dimensional shape information of the 2pCF down to scales as small as 5 h −1 Mpc it is expected that the AP test with redshift-space galaxy clustering anisotropy can be a powerful method to constraining the expansion history of the universe.

The redshift-space momentum power spectrum – I. Optimal estimation from peculiar velocity surveys

Abstract Low redshift surveys of galaxy peculiar velocities provide a wealth of cosmological information. We revisit the idea of extracting this information by directly measuring the redshift-space momentum power spectrum from such surveys. We provide a comprehensive theoretical and practical framework for estimating and fitting this from data, analogous to well-understood techniques used to measure the galaxy density power spectrum from redshift surveys. We formally derive a new estimator, which includes the effects of shot noise and survey geometry; we evaluate the variance of the estimator in the Gaussian regime; we compute the optimal weights for the estimator; we demonstrate that the measurements are Gaussian distributed, allowing for easy extraction of cosmological parameters; and we explore the effects of peculiar velocity (PV) measurement errors. We finish with a proof-of-concept using realistic mock galaxy catalogues, which demonstrates that we can measure and fit both the redshift-space galaxy density and momentum power spectra from PV surveys and that including the latter substantially improves our constraints on the growth rate of structure. We also provide theoretical descriptions for modelling the non-linear redshift-space density and momentum power spectrum multipoles, and forecasting the constraints on cosmological parameters using the Fisher information contained in these measurements for arbitrary weights. These may be useful for measurements of the galaxy density power spectrum even in the absence of peculiar velocities.

Neutrino mass and dark energy constraints from redshift-space distortions

Cosmology in the near future promises a measurement of the sum of neutrino masses ∑ mν, a fundamental Standard Model parameter, as well as substantially-improved constraints on the dark energy. We use the shape of the BOSS redshift-space galaxy power spectrum, in combination with CMB and supernova data, to constrain the neutrino masses and the dark energy. Essential to this calculation are several recent advances in non-linear cosmological perturbation theory, including fast Fourier transform methods, redshift space distortions, and scale-dependent growth. Our 95% confidence upper bound ∑ mν < 180 meV degrades substantially to ∑ mν < 540 meV when the dark energy equation of state and its first derivative are also allowed to vary, representing a significant challenge to current constraints. We also study the impact of additional galaxy bias parameters, finding that a greater allowed range of scale-dependent bias only slightly shifts the preferred ∑ mν, weakens its upper bound by ≈ 20%, and has a negligible effect on the other cosmological parameters.

Measurement of marked correlation functions in SDSS-III Baryon Oscillation Spectroscopic Survey using LOWZ galaxies in Data Release 12

Marked correlation functions, which are sensitive to the clustering of galaxies in different environments, have been proposed as constraints on modified gravity models. We present measurements of the marked correlation functions of galaxies in redshift space using 361,761 LOWZ ($z_{\rm eff} = 0.32$) galaxies from the Sloan Digital Sky Survey III (SDSS III) Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 (DR12) and compare them to $\Lambda$CDM+General Relativity simulations. We apply mass cuts to find the best match between the redshift space autocorrelation function of subhaloes in the simulation and in the observations. We then compare the marked correlation functions, finding no significant evidence for deviations of the marked correlation functions of LOWZ galaxies from $\Lambda$CDM on scales $6 \ h^{-1}$Mpc $ \leq s \leq$ $69 \ h^{-1}$Mpc. The constraining power of marked correlation functions in our analysis is limited by our ability to model the autocorrelation function of galaxies on small scales including the effect of redshift distortions. The statistical errors are well below the differences seen between marked correlation functions of $f$(R) gravity models and $\Lambda$CDM in recent publications (Armijo et al., Hern\'{a}ndez-Aguayo et al.) indicating that improved future theoretical analyses should be able to rule out some models definitively.

The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: a tomographic measurement of structure growth and expansion rate from anisotropic galaxy clustering in Fourier space

We perform a tomographic structure growth and expansion rate analysis using the monopole, quadrupole and hexadecapole of the redshift-space galaxy power spectrum derived from the Sloan Digital Sky Survey (SDSS-III) Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 combined sample, which covers the redshift range of $0.20<z<0.75$. By allowing for overlap between neighbouring redshift slices in order to extract information on the light-cone, we successfully obtain joint BAO and RSD constraints with a precision of $2-3\%$ for $D_A$, $3-10\%$ for $H$ and $9-12\%$ for $f\sigma_8$ with a redshift resolution of $\Delta z\sim0.04$. Our measurement is consistent with that presented in arXiv:1709.05173, where the analysis is performed in configuration space. We apply our measurement to constrain the $f(R)$ gravity model, and find that the 95\% CL upper limit of ${\rm log_{10}}B_0$ can be reduced by 11\% by our tomographic BAO and RSD measurement.

Geometrical compression: a new method to enhance the BOSS galaxy bispectrum monopole constraints

ABSTRACT We present a novel method to compress galaxy clustering three-point statistics and apply it to redshift space galaxy bispectrum monopole measurements from BOSS DR12 CMASS data considering a k-space range of $0.03-0.12\, h/\mathrm{Mpc}$. The method consists in binning together bispectra evaluated at sets of wavenumbers forming closed triangles with similar geometrical properties: the area, the cosine of the largest angle, and the ratio between the cosines of the remaining two angles. This enables us to increase the number of bispectrum measurements, for example by a factor of 23 over the standard binning (from 116 to 2734 triangles used), which is otherwise limited by the number of mock catalogues available to estimate the covariance matrix needed to derive parameter constraints. The $68{{\ \rm per\ cent}}$ credible intervals for the inferred parameters (b1, b2, f, σ8) are thus reduced by $\left(-39{{\ \rm per\ cent}},-49{{\ \rm per\ cent}},-29{{\ \rm per\ cent}},-22{{\ \rm per\ cent}}\right)$, respectively. We find very good agreement with the posteriors recently obtained by alternative maximal compression methods. This new method does not require the a-priori computation of the data vector covariance matrix and has the potential to be directly applicable to other three-point statistics (e.g. galaxy clustering, weak gravitational lensing, 21-cm emission line) measured from future surveys such as DESI, Euclid, PFS, and SKA.

A practical computational method for the anisotropic redshift-space three-point correlation function

We present an algorithm enabling computation of the anisotropic redshift-space galaxy 3-point correlation function (3PCF) scaling as $N^2$, with $N$ the number of galaxies. Our previous work showed how to compute the isotropic 3PCF with this scaling by expanding the radially-binned density field around each galaxy in the survey into spherical harmonics and combining these coefficients to form multipole moments. The $N^2$ scaling occurred because this approach never explicitly required the relative angle between a galaxy pair about the primary galaxy. Here we generalize this work, demonstrating that in the presence of azimuthally-symmetric anisotropy produced by redshift-space distortions (RSD) the 3PCF can be described by two triangle side lengths, two independent total angular momenta, and a spin. This basis for the anisotropic 3PCF allows its computation with negligible additional work over the isotropic 3PCF. We also present the covariance matrix of the anisotropic 3PCF measured in this basis. Our algorithm tracks the full 5-D redshift-space 3PCF, uses an accurate line of sight to each triplet, is exact in angle, and easily handles edge correction. It will enable use of the anisotropic large-scale 3PCF as a probe of RSD in current and upcoming large-scale redshift surveys.

Cosmological constraints from galaxy clustering in the presence of massive neutrinos

The clustering ratio is defined as the ratio between the correlation function and the variance of the smoothed overdensity field. In LCDM cosmologies not accounting for massive neutrinos, it has already been proved to be independent from bias and redshift space distortions on a range of linear scales. It therefore allows for a direct comparison of measurements (from galaxies in redshift space) to predictions (for matter in real space). In this paper we first extend the applicability of such properties of the clustering ratio to cosmologies that include massive neutrinos, by performing tests against simulated data. We then investigate the constraining power of the clustering ratio when cosmological parameters such as the total neutrino mass and the equation of state of dark energy are left free. We analyse the joint posterior distribution of the parameters that must satisfy, at the same time, the measurements of the galaxy clustering ratio in the SDSS DR12, and the angular power spectrum of temperature and polarization anisotropies of the CMB measured by the Planck satellite. We find the clustering ratio to be very sensitive to the CDM density parameter, but not very much so to the total neutrino mass. Lastly, we forecast the constraining power the clustering ratio will achieve with forthcoming surveys, predicting the amplitude of its errors in a Euclid-like galaxy survey. In this case, we find it is expected to improve the constraint at 95% level on the CDM density by 40% and on the total neutrino mass by 14%.

Maximal compression of the redshift-space galaxy power spectrum and bispectrum

We explore two methods of compressing the redshift space galaxy power spectrum and bispectrum with respect to a chosen set of cosmological parameters. Both methods involve reducing the dimension of the original data-vector ( e.g. 1000 elements ) to the number of cosmological parameters considered ( e.g. seven ) using the Karhunen-Lo\`eve algorithm. In the first case, we run MCMC sampling on the compressed data-vector in order to recover the one-dimensional (1D) and two-dimensional (2D) posterior distributions. The second option, approximately 2000 times faster, works by orthogonalising the parameter space through diagonalisation of the Fisher information matrix before the compression, obtaining the posterior distributions without the need of MCMC sampling. Using these methods for future spectroscopic redshift surveys like DESI, EUCLID and PFS would drastically reduce the number of simulations needed to compute accurate covariance matrices with minimal loss of constraining power. We consider a redshift bin of a DESI-like experiment. Using the power spectrum combined with the bispectrum as a data-vector, both compression methods on average recover the 68% credible regions to within 0.7% and 2% of those resulting from standard MCMC sampling respectively. These confidence intervals are also smaller than the ones obtained using only the power spectrum by (81%, 80%, 82%) respectively for the bias parameter b_1, the growth rate f and the scalar amplitude parameter A_s.

Properties and Origin of Galaxy Velocity Bias in the Illustris Simulation

Abstract We use hydrodynamical galaxy formation simulations from the Illustris suite to study the origin and properties of galaxy velocity bias, i.e., the difference between the velocity distributions of galaxies and dark matter inside halos. We find that galaxy velocity bias decreases with increasing ratio of galaxy stellar mass to host halo mass. In general, central galaxies are not at rest with respect to dark matter halos or the core of halos, with a velocity dispersion above 0.04 times that of the dark matter. The central galaxy velocity bias is found to be mostly caused by close interactions between the central and satellite galaxies. For satellite galaxies, the velocity bias is related to their dynamical and tidal evolution history after being accreted onto the host halos. It depends on the time after the accretion and their distances from the halo centers, with massive satellites generally moving more slowly than the dark matter. The results are in broad agreement with those inferred from modeling small-scale redshift-space galaxy clustering data, and the study can help improve models of redshift-space galaxy clustering.

Linear redshift space distortions for cosmic voids based on galaxies in redshift space

Cosmic voids found in galaxy surveys are defined based on the galaxy distribution in redshift space. We show that the large scale distribution of voids in redshift space traces the fluctuations in the dark matter density field $\stackrel{^}{\ensuremath{\delta}}(\mathbit{k})$ (in Fourier space with $\ensuremath{\mu}$ being the line-of-sight projected $\mathbit{k}$ vector), ${\stackrel{^}{\ensuremath{\delta}}}_{\mathrm{v}}^{s}(\mathbit{k})=(1+{\ensuremath{\beta}}_{\mathrm{v}}{\ensuremath{\mu}}^{2}){b}_{\mathrm{v}}^{s}\stackrel{^}{\ensuremath{\delta}}(\mathbit{k})$, with a beta factor that will be, in general, different than the one describing the distribution of galaxies. Only if voids are assumed to be quasilocal transformations of the linear (Gaussian) galaxy redshift space field does one get equal beta factors ${\ensuremath{\beta}}_{\mathrm{v}}={\ensuremath{\beta}}_{\mathrm{g}}=f/{b}_{\mathrm{g}}$ with $f$ being the growth rate and ${b}_{\mathrm{g}}$, ${b}_{\mathrm{v}}^{s}$ being the galaxy and void bias on large scales defined in redshift space. Indeed, in our mock void catalogs, we measure void beta factors in good agreement with the galaxy one. Further work needs to be done to confirm the level of accuracy of the beta factor equality between voids and galaxies, but in general the void beta factor needs to be considered as a free parameter for linear RSD studies.

Analytic halo approach to the bispectrum of galaxies in redshift space

We present an analytic formula for the galaxy bispectrum in redshift space on the basis of the halo approach description with the halo occupation distribution of central galaxies and satellite galaxies. This work is an extension of a previous work on the galaxy power spectrum, which illuminated the significant contribution of satellite galaxies to the higher multipole spectrum through the non-linear redshift space distortions of their random motions. Behaviors of the multipoles of the bispectrum are compared with results of numerical simulations assuming a halo occupation distribution of the LOWZ sample of the SDSS-III BOSS survey. Also presented are analytic approximate formulas for the multipoles of the bispectrum, which is useful to understanding their characteristic properties. We demonstrate that the Fingers of God effect is quite important for the higher multipoles of the bispectrum in redshift space, depending on the halo occupation distribution parameters.

The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: tomographic BAO analysis of DR12 combined sample in Fourier space

We perform a tomographic baryon acoustic oscillations (BAO) analysis using the monopole, quadrupole and hexadecapole of the redshift-space galaxy power spectrum measured from the pre-reconstructed combined galaxy sample of the completed Sloan Digital Sky Survey (SDSS-III) Baryon Oscillation Spectroscopic Survey (BOSS) Data Release (DR)12 covering the redshift range of $0.20<z<0.75$. By allowing for overlap between neighbouring redshift slices, we successfully obtained the isotropic and anisotropic BAO distance measurements within nine redshift slices to a precision of $1.5\%-3.4\%$ for $D_V/r_d$, $1.8\% -4.2\%$ for $D_A/r_d$ and $3.7\% - 7.5\%$ for $H \ r_d$, depending on effective redshifts. We provide our BAO measurement of $D_A/r_d$ and $H \ r_d$ with the full covariance matrix, which can be used for cosmological implications. Our measurements are consistent with those presented in \citet{Acacia}, in which the BAO distances are measured at three effective redshifts. We constrain dark energy parameters using our measurements, and find an improvement of the Figure-of-Merit of dark energy in general due to the temporal BAO information resolved. This paper is part of a set that analyses the final galaxy clustering dataset from BOSS.

Effect of relative velocity and density perturbations between baryons and dark matter on the clustering of galaxies

Pre-recombination acoustic oscillations induce non-adiabatic perturbations between baryons and dark matter, corresponding to a constant relative-density $\delta_{bc}$ and decaying relative-velocity perturbation $\vec{v}_{bc}$. Due to their significant large-scale correlations and prominent baryon acoustic oscillation (BAO) features, these modes are potentially important for the use of the BAO as standard ruler. We present a complete treatment of the effects of the baryon-CDM perturbations on galaxy clustering in the context of a rigorous perturbative bias expansion. The leading effects are proportional to $\delta_{bc}$ and $\theta_{bc} = \partial_i v_{bc}^i$. We estimate the magnitude of these terms through the excursion set approach. The contribution from $v_{bc}^2$, which has attracted significant attention recently, contributes at subleading (1-loop) order. The relative-density contribution $\delta_{bc}$ is expected to be by far the largest contribution. We also point out contributions to the galaxy velocity bias, the largest of which is simply $v_{bc}$, leading to a term $\propto \mu^2\theta_{bc}$ in the redshift-space galaxy power spectrum $P_g^s(k,\mu)$. Complete expressions of the galaxy power spectrum at 1-loop order are given, which contain several new terms.

Accurate and efficient halo-based galaxy clustering modelling with simulations

Small- and intermediate-scale galaxy clustering can be used to establish the galaxy-halo connection to study galaxy formation and evolution and to tighten constraints on cosmological parameters. With the increasing precision of galaxy clustering measurements from ongoing and forthcoming large galaxy surveys, accurate models are required to interpret the data and extract relevant information. We introduce a method based on high-resolution N-body simulations to accurately and efficiently model the galaxy two-point correlation functions (2PCFs) in projected and redshift spaces. The basic idea is to tabulate all information of haloes in the simulations necessary for computing the galaxy 2PCFs within the framework of halo occupation distribution or conditional luminosity function. It is equivalent to populating galaxies to dark matter haloes and using the mock 2PCF measurements as the model predictions. Besides the accurate 2PCF calculations, the method is also fast and therefore enables an efficient exploration of the parameter space. As an example of the method, we decompose the redshift-space galaxy 2PCF into different components based on the type of galaxy pairs and show the redshift-space distortion effect in each component. The generalizations and limitations of the method are discussed.

Gaussian covariance matrices for anisotropic galaxy clustering measurements

Measurements of the redshift-space galaxy clustering have been a prolific source of cosmological information in recent years. Accurate covariance estimates are an essential step for the validation of galaxy clustering models of the redshift-space two-point statistics. Usually, only a limited set of accurate N-body simulations is available. Thus, assessing the data covariance is not possible or only leads to a noisy estimate. Further, relying on simulated realisations of the survey data means that tests of the cosmology dependence of the covariance are expensive. With these points in mind, this work presents a simple theoretical model for the linear covariance of anisotropic galaxy clustering observations with synthetic catalogues. Considering the Legendre moments (`multipoles') of the two-point statistics and projections into wide bins of the line-of-sight parameter (`clustering wedges'), we describe the modelling of the covariance for these anisotropic clustering measurements for galaxy samples with a trivial geometry in the case of a Gaussian approximation of the clustering likelihood. As main result of this paper, we give the explicit formulae for Fourier and configuration space covariance matrices. To validate our model, we create synthetic HOD galaxy catalogues by populating the haloes of an ensemble of large-volume N-body simulations. Using linear and non-linear input power spectra, we find very good agreement between the model predictions and the measurements on the synthetic catalogues in the quasi-linear regime.

2MTF – V. Cosmography, β, and the residual bulk flow

Using the Tully-Fisher relation, we derive peculiar velocities for the 2MASS Tully-Fisher Survey and describe the velocity field of the nearby Universe. We use adaptive kernel smoothing to map the velocity field, and compare it to reconstructions based on the redshift space galaxy distributions of the 2MASS Redshift Survey (2MRS) and the IRAS Point Source Catalog Redshift Survey (PSCz). With a standard $\chi^2$ minimization fit to the models, we find that the PSCz model provides a better fit to the 2MTF velocity field data than does the 2MRS model, and provides a value of $\beta$ in greater agreement with literature values. However, when we subtract away the monopole deviation in the velocity zeropoint between data and model, the 2MRS model also produces a value of $\beta$ in agreement with literature values. We also calculate the `residual bulk flow': the component of the bulk flow not accounted for by the models. This is $\sim 250$ km/s when performing the standard fit, but drops to $\sim 150$ km/s for both models when the aforementioned monopole offset between data and models is removed. This smaller number is more in line with theoretical expectations, and suggests that the models largely account for the major structures in the nearby Universe responsible for the bulk velocity.

The VIMOS Public Extragalactic Redshift Survey

Aims. Using the VIMOS Public Extragalactic Redshift Survey (VIPERS) we aim to jointly estimate the key parameters that describe the galaxy density field and its spatial correlations in redshift space. Methods. We use the Bayesian formalism to jointly reconstruct the redshift-space galaxy density field, power spectrum, galaxy bias and galaxy luminosity function given the observations and survey selection function. The high-dimensional posterior distribution is explored using the Wiener filter within a Gibbs sampler. We validate the analysis using simulated catalogues and apply it to VIPERS data taking into consideration the inhomogeneous selection function. Results. We present joint constraints on the anisotropic power spectrum as well as the bias and number density of red and blue galaxy classes in luminosity and redshift bins as well as the measurement covariances of these quantities. We find that the inferred galaxy bias and number density parameters are strongly correlated although these are only weakly correlated with the galaxy power spectrum. The power spectrum and redshift-space distortion parameters are in agreement with previous VIPERS results with the value of the growth rate $f\sigma_8 = 0.38$ with 18% uncertainty at redshift 0.7.